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October  2019, 24(10): 5597-5600. doi: 10.3934/dcdsb.2019073

Minimal forward random point attractors need not exist

Michael Scheutzow

Technische Universität Berlin, Fak. Ⅱ, Institut für Mathematik, Sekr. MA 7-5, Straẞe des 17. Juni 136, Germany

Received  August 2018 Revised  November 2018 Published  October 2019 Early access  April 2019

It is well-known that random attractors of a random dynamical system are generally not unique. It was shown in [1] that if there exist more than one pullback or weak random attractor which attracts a given family of (possibly random) sets, then there exists a minimal (in the sense of smallest) one. This statement does not hold for forward random attractors. The same paper contains an example of a random dynamical system and a deterministic family of sets which has more than one forward attractor which attracts the given family but no minimal one. The question whether one can find an example which has multiple forward point attractors but no minimal one remained open. Here we provide such an example.

Keywords: Random attractor, random dynamical system, point attractor, forward attractor, weak attractor.
Mathematics Subject Classification: Primary: 60H25; Secondary: 37B25, 37C70, 37H10, 37H99, 60H10.
Citation: Michael Scheutzow. Minimal forward random point attractors need not exist. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5597-5600. doi: 10.3934/dcdsb.2019073
References:
[1]

H. Crauel and M. Scheutzow, Minimal random attractors, J. Differential Equations, 265 (2018), 702-718.  doi: 10.1016/j.jde.2018.03.011.  Google Scholar

show all references

References:
[1]

H. Crauel and M. Scheutzow, Minimal random attractors, J. Differential Equations, 265 (2018), 702-718.  doi: 10.1016/j.jde.2018.03.011.  Google Scholar

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